Solve x^2+3x-4x^2-5x+6x^2-2x-3

$$x ^ { 2 } + 3 x - 4 x ^ { 2 } - 5 x + 6 x ^ { 2 } - 2 x - 3$$

Evaluate

$$3x^{2}-4x-3$$

Solution Steps

Combine x^{2} and -4x^{2} to get -3x^{2}.

$$-3x^{2}+3x-5x+6x^{2}-2x-3$$

Combine 3x and -5x to get -2x.

$$-3x^{2}-2x+6x^{2}-2x-3$$

Combine -3x^{2} and 6x^{2} to get 3x^{2}.

$$3x^{2}-2x-2x-3$$

Combine -2x and -2x to get -4x.

$$3x^{2}-4x-3$$

Factor

$$3\left(x-\frac{2-\sqrt{13}}{3}\right)\left(x-\frac{\sqrt{13}+2}{3}\right)$$

Steps Using the Quadratic Formula

Combine x^{2} and -4x^{2} to get -3x^{2}.

$$factor(-3x^{2}+3x-5x+6x^{2}-2x-3)$$

Combine 3x and -5x to get -2x.

$$factor(-3x^{2}-2x+6x^{2}-2x-3)$$

Combine -3x^{2} and 6x^{2} to get 3x^{2}.

$$factor(3x^{2}-2x-2x-3)$$

Combine -2x and -2x to get -4x.

$$factor(3x^{2}-4x-3)$$

Quadratic polynomial can be factored using the transformation ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right), where x_{1} and x_{2} are the solutions of the quadratic equation ax^{2}+bx+c=0.

$$3x^{2}-4x-3=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$x=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\times 3\left(-3\right)}}{2\times 3}$$

Square -4.

$$x=\frac{-\left(-4\right)±\sqrt{16-4\times 3\left(-3\right)}}{2\times 3}$$

Multiply -4 times 3.

$$x=\frac{-\left(-4\right)±\sqrt{16-12\left(-3\right)}}{2\times 3}$$

Multiply -12 times -3.

$$x=\frac{-\left(-4\right)±\sqrt{16+36}}{2\times 3}$$

Add 16 to 36.

$$x=\frac{-\left(-4\right)±\sqrt{52}}{2\times 3}$$

Take the square root of 52.

$$x=\frac{-\left(-4\right)±2\sqrt{13}}{2\times 3}$$

The opposite of -4 is 4.

$$x=\frac{4±2\sqrt{13}}{2\times 3}$$

Multiply 2 times 3.

$$x=\frac{4±2\sqrt{13}}{6}$$

Now solve the equation x=\frac{4±2\sqrt{13}}{6} when ± is plus. Add 4 to 2\sqrt{13}.

$$x=\frac{2\sqrt{13}+4}{6}$$

Divide 4+2\sqrt{13} by 6.

$$x=\frac{\sqrt{13}+2}{3}$$

Now solve the equation x=\frac{4±2\sqrt{13}}{6} when ± is minus. Subtract 2\sqrt{13} from 4.

$$x=\frac{4-2\sqrt{13}}{6}$$

Divide 4-2\sqrt{13} by 6.

$$x=\frac{2-\sqrt{13}}{3}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{2+\sqrt{13}}{3} for x_{1} and \frac{2-\sqrt{13}}{3} for x_{2}.

$$3x^{2}-4x-3=3\left(x-\frac{\sqrt{13}+2}{3}\right)\left(x-\frac{2-\sqrt{13}}{3}\right)$$